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NASA TT F-15,955 

r ■■ 

H. Landstorfer, H. Liska, H. Meinke and B. Miiller 


t: .'-TROMAGNETIC wave fields (Kanner (Leo) 

A.-..ociat6s) 20 p HC $4.00 CSCL 2CN 


G3/07 50372 

Translation of "Energlestromung in elektromagnetlschen 
Wellenfeldern," Nachrichtentechnische Zeltschrlft 
(NTZ-Communicatlons Journal), Vol. 25, No. 5, 
1972, pp. 225-231 




■ xs. *«« 



H. Landstorfer, H. Llska, H. Melnke and B. Muller 

Preliminary Remark /225 * 

In his field equations. Maxwell chose to represent field 
behavior by means of field strengths. This undoubtedly was due 
to historical reasons. We are of the view that field strengths 
provide more cf a formal mathematical description of the state of 
the field, whereas the distribution, motion and conversion of 
energy (conversion of electrical energy Into magnetic energy and 
vice versa) represent the actual physical process. It appears 
possible that better Information concerning the behavior of energy 
may also result in an Improved understanding of electromagnetic 
processes. A better physical understanding would assist the 
engineer In all of those complex cases In which mathematical com- 
putation Is not feasible or Is no longer reasonable because of 
the high mathematical outlay. Unfortunately, a great many prac- 
tical applications involving electromagnetic waves belong among 
the noncalculable cases, due to the severe boundary conditions and 
have only been amenable to experimental research so far. 

1. Instantaneous Energy Flow 

The Poyntlng vector, called the vector of energy flow den- 
sity in the following [5a], Indicates the direction of energy flow, 
via its own direction, and energy transport per unit time, re- 
ferred to the unit of area, via its length. In time-variable 
fields, it is defined for the time being as the tlme-vai'lable 
vector s of instantaneous energy flow density 

* Nrmbers in the margin indicate pagination in the foreign text, 

e X h 


As shown in Pig. 1, it is perpendicular to the vector e of in- 
stantaneous eirctric field strength and to the vector h of 
instantaneous magnetic field strength. Its length is 

s = 

Ih 'COS a 


where a is the angle between the spatial vectors e and h. Our 
studies on instantaneous energy movement [4] have shown that very 
complex and difficultly understood energy behavior occurs even in 
relatively simple cases. Moreover, all time-variable field pro- 
cesses have the basic disadvantage that they can be represented 
only through a series of pictures, e.g. instantaneous patterns; 
the result actually becomes completely recognizable only as 
a motion picture. The presentation of such a sequence of pictures 
in a book or Journal is expensive and yet not very satisfactory, 
ihe reader is referred to our instantaneous patterns of the near 
field of an antenna in [1] and [2], which are considerably 
clearer in effect in a motion picture which has been made. 

2. Time-Averaged Energy Flow 

Plotting the vector of time-averaged energy flow density, 
which thus indicates the mean of energy transport over time at a 
given location in terms cf direction and magnitude, has proven to 
be particularly simple and informative. This method was intro- 
duced by Wolter [3]. At each point in the field, there is thus 
a vector, constant over time, which indicates the direction of 
mean energy flow and its mean magnitude at this point. Actual 
energy movement is the result of superimposing an oscillation 
over this mean movement; cf. Section 3- We can then t-how stream- 
lines for time-averagtd energy flow in this space on the basis of 
known principles of flow theory; the vectors of time-averaged 
energy flow density are tangents to the streamlines. Such a 


Fig. 1. Vector of 
energy flow density 
(Poyntlng vector). 

streamline pattern Is very similar to 
stationary flow In a source-free fluid. 

We obtain time-averaged energy 
transport by means of a generalized 
pattern of effective values with the 
aid of complex amplitudes. In a rectan- 
gular coordinate system (Fig. 2) , complex 
electric field strength vector E has the 
complex components 

lz = Ex-ei^'; £y = £^-eiv.; £, = £,• el»". ( 3) 

Complex magnetic field strength 
vector H has the complex components 

Fig. 2. Vectors of a TM 

Hr = Hx- eif; Hy= Hy- eJ^«; H^^ H^- e'". . ( 4 ) 

\ As an expansion of equation (1), 

we can define a complex vector for energy flow density [6] 

S=SR + jSi = y(ExH*) 


H* is the complex conjugate vector of magnetic field strength. /226 
This definition of energy flow density is a generalization of the 
formula for complex apparent power, in which ono of the two co- 
efficients also appears with its complex conjugate value. The 
coefficient 1/2 always occurs in effective values when field 
strengths are given in peak values. 

The transport of real power is described by the real part, 
S|^. Sj^ is the actual energy flow density, which also establishes 


the direction of streamlines. The direction of Sr is the mean 
propagation direction of the energy. The imaginary part Sj from 
equation (5) represen;s a reactive power density and describes an 
oscillating movement of energy about its time-averaged movement 
(see Section 3) . 

The formula for real part Sr Is relatively extensive for 
general three-dimensional wave processes, so two simpler examples 
will merely be considered formally in the following. Fig. 2 shows 
the first case. Here, the electric fields of the wave lie in an 
x-z plane, and magnetic field Hy is perpendicular to the plane 
(TM wave). Fig. 3 shows the second case, in which magnetic 
field lies in a plane with components H^^ and H^;, and electric 
field is perpendicular to this plane as an Ey component (TE 
wave). From the convention in Fig. 1, with a = 90°, components 
E;c and Hy from equations (3) and {k) form the component Srz of 
time-averaged energy flow density in Fig. 2: 

-Sr, — -^ExUy- cos (v»z — tpv) 


and components Eg and Hy form the component Sr^ of energy flow; 

SVLX = —-^EtHy- COS (y, - <py) 


Pig. 3. Vectors of a TE 

The (mathematically) positive 
directions of the components of 
field strength and energy flow den- 
sity are diagrammed in Fig. 2. If 
we apply the convention in Fig. 1 
to component Sr^, then, according 
to equation (7), Srx must be shown 
in the negative direction. 



In Fig. 3, components Ey and H^^ form the component S^^ of 
energy flow: 

likewise, components Ey and Hg form the component Srx of energy 

1 (Q) 

In these two special cases, the energy streamlines lie in 
the plane given and thus can be represented graphically by simple 
means. With the exception of a three-dimensional case in Fig. 13> 
only planar cases will be represented in the following. As is 
conventional for planar flows, the streamlines are drawn at dis- 
tances such that equal energy flows move between each pair of 
adjacent streamlines. Thus the distance between streamlines in 
such pictures is a measure of flow density. A small interval 
indicates high flow density. 

3. Ellipse of the Instantaneous Vector of Energy Flow 

It is shown in [7] that in the special cases shown in Figs. 
2 and 3, instantaneous vector s(t) in equation (1) is obtained 
from the time-averaged value Sr in equations (6) through (9) with 
an additional time-dependent vector AsCt): 

8(t) - Sj^ + A8(t). (10) 

This As follows an ellipse with time, as shown in Fig. i|, 
in such a manner that 8 becomes zero at a certain time. Sfj and 
Si are conjugate diameters of the ellipse, and the ellipse can 
be constructed from these two vectors by known methods. In a 

plane wave with linear propagation. 

•(0 = Se[1 +coa(«>< + i)]. 


and the ellipse degenerates Into a 
straight line which has the direction of 
Sr. The ellipse thus describes the 
periodic oscillations superimposed on 
time-averaged energy flow. In a cer- 
tain way, the transverse oscillations 
represent a reactive power in the field. 
We suspect that this reactive power has an 
effect on the behaviors of streamlines for real power. 

Fig. 4. Ellipse of the 
complex vector of 
energy flow density. 

4 . Energy Velocity 

If we wish to characterize the movement of energy in a wave 
field in detail, we must define an energy velocity. In a plane 
TEM wave, linear movement occurs everywhere and at all times and 
is at the velocity of light. In a more general wave field, energy /227 
velocity , in terms of direction and magnitude, is a function of 
position and time. This applies to waves in homogeneous 

The energy dW in an infinitesimal volume element dV will be 
called an energy packet. The energy packet moves at the in- 
stantaneous energy velocity associated with the particular location 
and time. A definition of velocity is not without problems, in 
that the identity of the energy packet as an individual cannot be 
determined along its path by measurement. A definition of 
velocity for unidentifiable individual entitles is always of a 
purely formal nature, but must be free of contradiction, i.e. 
must be compatible with all measured physical phenomena. In 
spite of thJs difficulty, we believe that the energy velocity 
defined by us can provide certain presentational advantages and 

makes certain verifiable conclusions possible, and thus is cer- 
tainly more than a pure formalism. 

A noncontradictory definition for a velocity vector is ob- 
tained if, in analogy to fluids, we define velocity as the 
quotient of energy flow density and energy density [4], If we 
use instantaneous values s for flow density and w for energy 
density here, we obtain the vector v of instantaneous energy 



If we use the time-averaged values Spj for flow density and 
Wjjj for energy density, we obtain mean energy velocity 





With the aid of instantaneous velocity v, we can calculate the 
path covered by an energy packet with time (instantaneous path; 
cf. Figs. 6, 7 and 9). With the aid of mean velocity, we obtain 
the mean path of the energy packet along the mean streamline, 
since vector Vm always has the direction of Sp. 

5. Homogeneous and Inhomogeneous Waves between Conducting Planes 

Pig. 5 shows a plane TEM wave between conducting planes. 
The case is that shown in Pig. 2, with £2*0. Only Ex exists. 
The electric flux lines are parallel vertical lines in direction 
E. Since the magnetic flux lines are perpendicular to the plane 
of the drawing, as in Fig. 2, energy flow has only a z-component. 
The horizontal lines marked with arrows are the instantaneous and 
time-averaged streamlines. The streamlines have the same inter- 
val between one another, and mean flow density is the same across 
the entire waveguide cross section. 













Pig. 5. Energy stream- 
lines in a TEM wave. 





Path of an 

^ l energy packet 
Y^ Streamline 

Pig. 6. TM wave between parallel 
conducting plane, a) Instanta- 
neous electric flux lines, b) 
Instantaneous and time-averaged 
paths of an energy packet, 
c) Mean energy flow and mean 
energy velocity. 

Fig. 6a shows the instantaneous 
electric flux lines of a TM wave be- 
tween parallel conducting planes [8]. 
This is a case of the type shown in 
Pig. 2 with time-dependent Oj^ and eg. 
Prom Fig. 1, with a » 90°, instan- 
taneous energy flow s is perpendicular 
to the instantaneous electric lines 

of flux. Since the lines 
of flux move, magnitude 
and direction of energy 
flow density are con- 
tinually changing. In- 
stantaneous velocity has 
components in the x- 
direction and z-direction. 
As an example. Fig, 6b 
shows a characteristic 
path of an energy packet 
which makes not only a 
longitudinal movement but 
also tranverse oscillations. 
Energy movement is sym- 
metrical with respect to 
the dashed center line, 
along which Sr ■ at all 
times and across which no 
energy therefore travels. 
Time-averaged energy flow 
has only a z-component, 
and the streamlines are 
The distribution of time- 


straight lines in the z-direction. 

averaged energy flow over the cross section is not uniform, as the 

distance between adjacent energy streamlines shows in Fig. 6c. 


A concentration of energy flow density occurs close to the bounding 
conducting planes. Tlme-aver£!,ed energy velocity Is a function of 
X. Fig. 6c shows the distribution of energy velocity Vj^g, 
defined by equation (13) > over the cross section on the right. 

Fig. 7 shows the Instantaneous magnetic flux lines of a TE 
wave between parallel conducting planes. This Is a case like 
that shown In Pig. 3» with time-variable hx and hg: 

e» = £„(x)-cos(a>t-/Jz); ^^^^ 

hx = Hg{x)'Cos(<ut-Pz); (15) 

hM'=H,(x)-sm{u)t-fiz). (16) 

Instantaneous energy flow s Is perpendicular to the In- 
stantaneous magnetic flux lines. It has the components 

«r = -eyh,'=:-^EyH^[i + co82(to«-/?z)]., (18) 

Sx varies at twice the freq ency but vanishes In the average over 
time. The component of Instantaneous energy flow s^ contains not 
only the alternating part cos 2(u)t - gz) but also a part Inde- 
pendent of time. Identical to the time average of energy flow S^^. 
Energy density w can likewise be resolved Into a DC part and an 
AC part, the DC part being Identical to mean energy w^: 


I Instantaneous velocity has components vx In the x-dlrectlon 
(transverse oscillation) and v^ In the z-dlrectlon (longitudinal 



1 Streamline 
•^M/Path of an 
-- ^energy packet 



movement). As an ex- 
ample, Pig. 7b shows th3 
characterJ'^* •'" path of 
an erer-"/ paei- - with 
transvei-^e oscillation. 
b) Energy movement la sym- 

metrical wltii respect 
to the dashed center 
line, on which S^^^ ■ 0, 
I.e. only Srz exists. 
Time-averaged energy 
flow has only a z- 
component. The stream- 
lines are straight lines 
In the z-dlrectlon. 
Time averaged energy flow 
Is distributed non- 
unlformly over the cross 
section, as the distance 
between adjacent stream- 
lines In Fig. 7c shows. 
A concentration of 
energy flow density occurs at the center of the waveguide. Fig. 
7c shows the distribution of mean energy velocity v„2 over the 
cross section on the right. The nonuniform distribution of energy 
velocity explains the familiar problems Involved in transmitting 
modulated signals with waveguides. Signals appear at the output 
of the waveguide with distorted modulation because they move 
faster at the inside, on the average, than at the outside. This 
effect is also referred to as the dispersion of a waveguide. 

Fig. 8 shows the ellipses of instantaneous flow density for 
a cross-sectional plane of the waveguide as an example for 
Pig. ^. Along the center line, the ellipse degenerates into a 
straight line, since Sji^^ " 0. 

Pig. 7. I'E wave between parallel con- 
ducting planes, a) Instantaneous 
magnetic flux lines, b) Instantaneous 
and time-averaged paths of an energy 
packet; c) Mean energy flow and mean 
energy velocity. 



— ^ 

Pig. 8. Ellipses of 
the complex vector 
of energy flow den- 
sity (Pig. H) In a 
TE wave between 
parallel conducting 
planes (Pig. 7). 

Fig. 9. Streamlines 3n a 
waveguide. Example of £;r. 
stantaneous energy path. 

Pig. 9 show the mean streamlines 
In an elbow and» In the form of a dashed 
line, an example of the Instantaneous 
path of an energy packet C^]. In the 
curved region, energy is "pushed" out- 
ward somewhat, on the average (by a 
sort of centrifugal force), and the in- 
stantaneous movement exhibits certain 
transverse oscillations there. 

Pig. 10a shows mean streamlines 
where a waveguide of parallel conducting 
planes widens abruptly, at a relatively 
low frequency, at which 
the higher modes are 
still quite aperiodic in 
inhomogeneity. Excitation 
is produced by a TEM wave 
arriving from the t. 
E is the direct f 
electric flux ] in the 
hcmogeneous regi-n. The 
energy flow behaves almost 
as in the potential flow of 
an incompressible fluid. 


Fig. 101-) shows the mean streamlines at a higher frequency, 
at which a low-energy region develops at the lower wall. The 
upper corner apparently has a suction effect at higher frequencies, 
so energy flows upward there. The resultant low-energy region on 
the lower conductor draws the energy downward again, and thus 
transverse oscillations arise in the energy flow, not only trans- 
verse oscillations In Instantaneous energy flow, as in Fig. 9» 
but also transverse oscillations in time-averaged energy flow. 





Pig. 10 Mean energy streamlines In a strip line 
which abruptly widens to width b. a) At a 
relatively low frequency (2b/> « 0.5), b) with a 
low-energy regie" C'b/Xo = 0.93), c) with an 
energy vortex (2b/X ^ O.96), d) with a multiple 
vortex (2b/X - 0-99). 

Fig. lOc shows an even higher frequency at which an energy 
vortex has developed [9]. Such vortices accompany very pronounced 
transverse oscillations in energy. These transverse oscillations 
can produce a series of sequential vortices. 

Fig. lOd shows energy streamlines in a multiple vortex. The 
vortices fill out each of the spaces which are freed of advancing 
energy by the transverse oscillations. A constant quantity of 
energy rotates in the vortex region, when averaged over time. 



In such complex wave fields, the Instantaneous movements are 
already becoming so difficult to follows [4] that they no longer 
provide any information. The mean streamlines, on the other hand, 
provide information on the behavior of inhomogeneity . For ex- 
ample, the vortex in Fig. 10c blocks the passage of energy in 
the region of the vortex, thereby reducing the effective cross 
section of the waveguide and lengthening the path of transported 
energy. The vortex can easily be detected with instruments 
through its effect on the inhomogeneity r'^f lection factor; see 
also [9]. 

6. Near Field of Rod Antennas 

Pig. 11 shows energy flow for a short rod antenna which is 
standing on a conducting plane and is fed coaxially. The cy- 
llndrically syr.unetrical fields have the character shown in 
Fig. 2. The fields were calculated in [7] and [10]. While con- 
ventional theories assume unrealistic slot feed, realistic feed 
through a coaxial line of finite size is treated here for the 
first time in an exact theory. Only in this way does it become 
possible to show how energy let-.-es the coaxial line into space 
along the streamlines. Pig. 11a shows the distribution of time- 
average vector Sr of energy flow density, and Fig. lib shows the 
associated streamlines [12]. It can be see.i here that the rod 
antenna does not "radiate"; rather, the energy moving from bottom 
to top in the coaxial feed line is altered in its direction of 
movement by the energy distribution that builds up in the 
antenna's near field and is distributed within the surrounding 
space. This change in streauline direction results from trans- 
verse energy oscillations, visible only in the instantaneous 
movement of energy, but still not in the mean streamlines in the 
case of short rods. These instantaneous transverse oscillations, 
which represent a type of reactive power in the field, can be 
seen from the imaginary part Si of energy flow density from 
equation (5) and the ellipses of complex energy flow density. 


f •-■' 


^» ^* \ 


/ ^ 


Fig. 11. Mean energy flow of a 
short rod antenna of length VlO. 
a) Distribution of the vector Sr 
of time-averaged energy flow 
density; b) Energy streamlines 
associated with Fig. 11a; 
c) Ellipses of tiie complex 
energy flow density vector along 
three energy streamlines from 
Pig. lib. 


As an example. Pig. lie 
shows these ellipses along 
three streamlines from 
Pig. lib. The dimensions of 
the ellipses perpendicular 
to the streamlines are a 
good measure of the magnitude 
of transverse oscilllations. 
The ellipses become flatter 
and the oscillations 
smaller with increasing 
distance from the antenna. 

If the rod is made 
longer, the transverse os- 
cillations become larger 
and also visible on the 
mean streamlines. Fig. 12 
shows the streamlines in 
the region around a longer 
rod. Once the mean stream- 
lines oscillate visibly 
back and forth, as in Fig. 
12, the far field pattern /230 
moves away from the familiar 
vertical pattern of the 
short radiator. Pronounced 
transverse oscillations 
produce a lobing of the 
far field pattern. In the 
future it should be pos- 
sible to explain the 
behavior of a wave in the 
far field quantitatively. 

too, on the basis of 
transverse energy os- 
cillations in the 
antenna's near field. 

7. Effective Area of 
Receiving Antennas 

Fig. 12. Mean energy streamlines of 

a rod antenna of length X. The near field of 

receiving antennas has 
been calculated in [11]. Representing the field with flux lines 
proves to be problematical and unclear here, since the flux 
lines require a time-variable three-dimensional mode of repre- 
sentation (superposition of a plane and a cylindrically symmetri- 
cal wave field). On the other hand, it is again simple and 
informative to describe the reception process with streamlines 
of time-averaged energy flow in this case [13]. 

Fig. 13 shows a short rod antenna above a conducting plane, 
acting as a receiving antenna. A complex resistance Z_ equal to 
the complex conjugate Impedance of the rod (internal resistance of 
source) exists between the base of the rod antenna and the con- 
ducting plane, acting as the dlssipator of received power. It is 
that dlssipator which draws the maximum possible (available) real 
power from the receiving antenna. A plane TEM wave passes along 
the conducting plane, from right to left in Fig. 13, with elec- 
tric field perpendicular to the conducting plane. Fig. 13a shows 
the distribution of the vector of time-averaged energy flow den- 
sity in a vertical plane through the radiator. From these 
vectors we obtain the streamlines shown in the vertical plane in 
Fig. 13b. Pig. 13b also shows the streamlines in the horizontal 
conducting plane in perspective. The linear intersection of the 
two planes shown is the direction of the arriving wave. There 
are boundary streamlines which divide the wave field into two 
regions. Two such boundaries are shown with dashed lines 





• •rs-''^' ~ 

^ -^ • • /// / ^ ^ 

' ' ' '/' 1 1 1 

' ' ' / ' M \ 
' ' ' ," ^ \ \ \ 

' 1 t f ^ ^ ^ " 

/ / / / • X ^ ^ 

y y y y '^ " ^ - 
y'" — -" 

H luijii: 

Pig. 13. Mean energy flow of a short 
receiving antenna, a) Distribution 
of the vector Sr of time-averaged 
energy flow density in a vertical 
plane through the radiator. The 
boundary streamline is dashed in. 
b) Mean energy streamlines (the 
horizontal plane is shown in per- 
spective), c) Perspective repre- 
sentation of boundary streamlines. 

In the vertical plane in 
Fig. 13b, all those 
streamlines move into 
the base of the antenna 
which are located under 
the bo^undary. All 
streamlines above the 
boundary go past the 
antenna. The boundary 
thus separates that 
part of the wave which 
passes into the re- 
ceiving antenna. Like- 
wise Pig. 13b shows 
the dashed boundary in 
the horizontal plane. 
All of the boundary 
streamlines are shown 
together in perspective 
in Fig. 130. They ter- 
minate at point P, 
perpendicular to the 
line which passes through 
the base of the antenna 
and is in the direction 
of the arriving wave, 
and they delimit a 
certain space. All 
streamlines outside this 
space flow past the 
receiving antenna. The 
streamlines within the 
delimited area all 
terminate at the base 


of the receiving antenna in dlssipator Z. The energy transported 
along these streamlines yields the received power. The boundary 
streamlines asymptotically approach straight lines parallel to the 
direction of the wave with increasing distance from the receiving 
antenna. The area delimited by them asymptotically approachs a 
cross section in the process which represents the effective area 
of the antenna , for the energy of the arriving plane wave passing 
through this area is absorbed by a receiving antenna with optimum 
dissipator Z, and this is the definition of effective area. We 
obtain Information regarding the shape of this effective area for 
the first time in this manner. The size of the effective area 
produced in this manner is equal to the values obtained theo- 
reticallyt to several percent, within the range of feasible 
numerical precision, e.g. 3X /16it for short rods. Effective area 
has been calculated for various rod lengths. The height of the 
effective area increases with increasing rod height and is greater 
than rod height. Its width decreases with increasing rod height 

The possibility of calculating the shape of the effective 
area is an interesting indication of the extensive new information 
about complex electromagnetic wave fields which can be obtained 
by calculating energy streamlines. The system of streamlines can 
be supplemented with a family of orthogonal curves, which we 
call "phase lines." These phase lines describe the time taken by 
the wave process along the streamlines. 



1. Llska, H., Meinke, H., and Mohr, C, "Wave separation from a 

conical antenna," Nachrlchtentech. Z. 23, 7^-79 (1970). 

2. Landstorfer, F. , "Wave separation from rod antennas," 

Nachrlchtentech. Z. 23. 273-278 (1970). 

3. Wolter, H., "The problem of the path of light In total 

reflection," Z. Naturforsch. 5A. 276-283 (1950). 

4. Llska, H. , "Energy transport In electromagnetic wave fields," 

dissertation, Munich Engineering University, 1970. 

5. Macke, W., Elektromagnetlsche Felder [Electromagnetic Fields], 

Section 13^, "The streamline pattern," 3rd edition, 
Leipzig, 1965. 

5a. Ibid. , Section 6l4. 

6. Meinke, H.H. and Gundlach, F.W., Taschenbuch der Hoch- 

frequenztechnlk [Handbook of Radio Frequency Engineering], 
Section H^l, 3rd edition, Berlin, Sprlnger-Verlag, 1968. 

7. Landstorfer, F. , "Wave separation and energy transport in the 

near fields of rod antennas used for transmission," 
habilltatlon paper, Munich Engineering University, 1971. 

8. Meinke, H., Einfuhrung in die Elektrotechnik hoherer Frequenzen 

[Introduction to Radio Frequency Electrical Engineering], 
Section II. 1, Vol. 2, 2nd edition, Berlin, Sprlnger-Verlag, 

9. Llska, H. , " Experimental evidence of the two elementary 

types of energy vortices in wave fields," Nachrlchtentech. 
Z^ 23, 445-^'^8 (1970). 

10. Landstorfer, F. , "Admittance and current distribution of 

linear cylindrical rod antennas," Arch, elektr. Ubertr. 23, 
61-69 (1969). 

11. Miiller, B., "Calculating the electromagnetic field about a 

receiving antenna," Arch. Elektron. t)bertr.-techn. 26 
(1972) (In press). 

12. Landstorfer, F., "Energy transjjort in the near field of rod 

antennas," Arch. Elektron. Ubertr. -techn. 26 (1972) 
(in press) . 

13. Miiller, B., "Calculating energy flow ln..the near field of a 

receiving antenna," Arch. Elektron. Ubertr. -techn. 26 (1972) 
(In press) .